The paper discusses the design of idealized tropical cyclone experiments in atmospheric general circulation models (AGCMs). The evolution of an initially weak, warm-core vortex is investigated over a 10-day period with varying initial conditions that include variations of the maximum wind speed and radius of maximum wind. The initialization of the vortex is built upon prescribed 3D moisture, pressure, temperature, and velocity fields that are embedded into tropical environmental conditions. The initial fields are in exact hydrostatic and gradient-wind balance in an axisymmetric form. The formulation is then generalized to provide analytic initial conditions for an approximately balanced vortex in AGCMs with height-based vertical coordinates. An extension for global models with pressure-based vertical coordinates is presented. The analytic initialization technique can easily be implemented on any AGCM computational grid.
The characteristics of the idealized tropical cyclone experiments are illustrated in high-resolution model simulations with the Community Atmosphere Model version 3.1 (CAM 3.1) developed at the National Center for Atmospheric Research. The finite-volume dynamical core in CAM 3.1 with 26 vertical levels is used, and utilizes an aquaplanet configuration with constant sea surface temperatures of 29°C. The impact of varying initial conditions and horizontal resolutions on the evolution of the tropical cyclone–like vortex is investigated. Identical physical parameterizations with a constant parameter set are used at all horizontal resolutions. The sensitivity studies reveal that the initial wind speed and radius of maximum wind need to lie above a threshold to support the intensification of the analytic initial vortex at horizontal grid spacings of 0.5° and 0.25° (or 55 and 28 km in the equatorial regions). The thresholds lie between 15 and 20 m s−1 with a radius of maximum wind of about 200–250 km. In addition, a convergence study with the grid spacings 1.0°, 0.5°, 0.25°, and 0.125° (or 111, 55, 28, and 14 km) shows that the cyclone gets more intense and compact with increasing horizontal resolution. The 0.5°, 0.25°, and 0.125° simulations exhibit many tropical cyclone–like characteristics such as a warm-core, low-level wind maxima, a slanted eyewall-like vertical structure and a relatively calm eye. The 0.125° simulation even starts to resolve spiral rainbands and reaches maximum wind speeds of about 72–83 m s−1 at low levels. These wind speeds are equivalent to a category-5 tropical cyclone on the Saffir–Simpson hurricane scale. It is suggested that the vortex initialization technique can be used as an idealized tool to study the impact of varying resolutions, physical parameterizations, and numerical schemes on the simulation and representation of tropical cyclone–like vortices in global atmospheric models.
Numerical modeling of tropical cyclones in general circulation models (GCMs) presents several challenges, including the relatively small size of the storms, intense convection and the interaction of large-scale and small-scale processes. A typical tropical cyclone has a radius of maximum wind (RMW) on the order of 10–100 km (Emanuel 2003), which is mostly unresolved at typical climate model resolutions of about 100–200 km. Despite this limitation, early GCM studies since the 1980s and early 1990s have succeeded in simulating tropical low pressure systems that have many tropical cyclone–like characteristics, such as a warm-core structure and realistic regions of formation. These findings are summarized in Walsh (2008) who also provides an overview of the current state-of-the-art of tropical cyclone modeling with climate models.
Tropical cyclone modeling in GCMs is a lively research field since modern computing architectures now allow very high horizontal resolutions that even approach the transition to nonhydrostatic scales. In particular, over the last 5 years numerous studies have discussed the generation and development of tropical cyclones with a variety of high-resolution GCMs. For example, Atlas et al. (2005) and Shen et al. (2006a,b) demonstrated the ability of the National Aeronautics and Space Administration (NASA) hydrostatic finite-volume GCM to simulate tropical cyclones successfully at horizontal resolutions of 0.25° and 0.125°, or 28 km and 14 km in equatorial regions, respectively. Another example is the study by Oouchi et al. (2006) who used a 20-km high-resolution global atmospheric model developed by the Meteorological Research Institute and Japan Meteorological Agency (MRI/JMA). They simulated the frequency, distribution, and intensity of tropical cyclones in the current climate, and despite some shortcomings the model was successful in reproducing the overall geographical distribution and frequency of tropical storms. However, similar to Shen et al. (2006b), Oouchi et al. (2006) had difficulty simulating the maximum intensity, suggesting that even higher resolutions may be required to simulate both tropical cyclone tracks and intensity accurately. Other recent investigations include the studies by Bengtsson et al. (2007) and Zhao et al. (2009), who evaluated tropical cyclone statistics in the Max-Planck Institute for Meteorology ECHAM5 model at the spectral resolution T319 (approximately 42 km) and in the Geophysical Fluid Dynamics Laboratory Atmospheric Model version 2.1 (AM2.1) on a 0.5° grid, respectively. The research shows that GCMs have developed an increasingly sophisticated ability to simulate tropical cyclones. This is further supported by the tropical cyclone assessments in the latest Intergovernmental Panel on Climate Change (IPCC) report that highlights the potential of recent GCMs to model tropical cyclones (Randall et al. 2007). Most recently, global cloud-resolving models, such as the Nonhydrostatic Icosahedral Atmospheric Model (NICAM), have shown enhanced skill in the simulation of real and projected future tropical cyclone activity (Fudeyasu et al. 2008; Yamada et al. 2010).
A number of recent studies have used simplified models to understand the factors that influence tropical cyclogenesis and rapid intensification. For example, Nolan (2007) used the Weather Research and Forecast model (WRF) developed at the National Center for Atmospheric Research (NCAR) at a grid spacing of 6 km with a 2-km nested grid to investigate the development and structure of tropical cyclones on a constant f plane. The model simulations started from prescribed initial conditions that favor cyclogenesis. In particular, the initial conditions were based on a tropical sounding with no mean wind and no wind shear, and the sea surface temperature (SST) was held constant at 29°C. In addition, Nolan et al. (2007) and Nolan and Rappin (2008) used WRF at high-resolutions (4 km) to investigate the impact of environmental variables, such as the SST, Coriolis parameter, mean surface wind, and wind shear on the evolution and intensification of a preexisting, weak, warm-core vortex into a tropical cyclone on an f plane. Similarly, Hill and Lackmann (2009) studied the development of an initial vortex in hydrostatic and gradient–wind balance in a warm moist environment to investigate the impact of the grid spacing, turbulence parameterizations, and surface layer fluxes in WRF. Such investigations have been very successful in simulating the development from an initially weak vortex to a hurricane-strength tropical cyclone, and shed light on cyclogenesis processes and the cyclone structure. The findings suggest that the lessons learned in idealized simulations are also relevant and applicable to real conditions. The idealized simulations are thereby a modeling tool that allow for further studies with respect to the horizontal and vertical model resolutions, physics parameterizations, and even the choice of the GCM dynamical core.
Inspired by the success of tropical cyclone simulations in regional models like WRF, we show that similar evolutions of an initially weak vortex into a tropical cyclone–like vortex can also be simulated in high-resolution GCMs that employ grid spacings of 0.5° and finer. This paper describes the design of the suggested initial conditions, the chosen aquaplanet setup of the GCM with constant SSTs and sensitivity tests. The first goal of this paper is to introduce a set of analytic initial conditions to initialize a weak, warm-core vortex in an aquaplanet configuration of NCAR’s Community Atmosphere Model version 3.1 (CAM 3.1) at high horizontal resolutions. The second goal is to exemplify the use of the analytic initial conditions by exploring the sensitivity of GCM simulations to varying initial conditions that include variations of the maximum wind speed and radius of maximum wind. From such sensitivity tests a control case is determined that simulates the development of the initial vortex into an intense tropical cyclone–like vortex. The control case can then be used to explore the model sensitivity to horizontal resolution. In general, the hope is that increased resolutions in GCMs in combination with adequate physical parameterizations will improve future climate simulations to a point that they can be reliably used to study the impact of changing climatic conditions on tropical cyclone statistics.
The paper is organized as follows. Section 2 discusses the design of the analytic initial conditions for the idealized tropical cyclone experiments. The initial setup comprises tropical environmental conditions with an embedded vortex in hydrostatic and axisymmetric gradient–wind balance. Section 3 provides a brief description of the NCAR model CAM 3.1. Section 4 reviews the results of the simulations in aquaplanet mode with different initial conditions for two different model resolutions of 0.5° and 0.25°. The identified control case is then used in section 5 to assess the evolution of the highly energetic idealized cyclone at the four horizontal resolutions: 1.0°, 0.5°, 0.25°, and 0.125°. This includes a discussion of the simulated convective and large-scale precipitation, moisture, temperature, and vertical velocity fields. The conclusions and summary are provided in section 6, as well as an outline of future work.
2. Idealized initial conditions
The first goal is to define an analytic set of idealized initial conditions that favor the development of tropical cyclone–like vortices in GCMs over the course of 10 simulation days. Our initialization technique differs from traditional bogussing techniques, as for example described in Leslie and Holland (1995) or Wang et al. (2008), since we prescribe analytic functions for the initial data that can readily be evaluated on any computational grid. The section is organized as follows. First, the 1D vertical profiles of the background temperature, moisture, and pressure conditions are introduced. Second, we use an axisymmetric approach to define a vortex in hydrostatic and gradient–wind balance. Third, we project the axisymmetric conditions onto the spherical domain for height-based vertical coordinate systems, and fourth, we generalize the formulation for pressure-based vertical coordinates that are most often used in GCMs today. The latter part involves simple fixed-point iterations that can also be applied to isentropic vertical coordinates [see the appendix of Jablonowski and Williamson (2006a)]. All parameters and physical constants used in the derivation are listed in Table 1.
a. 1D vertical profiles for the background conditions
The first step is to provide analytic background moisture and virtual temperature profiles that fit the observed tropical soundings by Jordan (1958). The analytic form of the background specific humidity profile q(z) as a function of height z is specified as
where zt = 15 km approximates the tropopause height as seen in the Jordan (1958) sounding; q0 is the specific humidity at the surface (z = 0 m), which is taken to be 21 g kg−1; and qt is the specific humidity in the upper atmosphere set to a constant value of 10−8 g kg−1. The specific humidity q0 was chosen to match the value of the relative humidity at the surface from the Jordan (1958) sounding, using a surface temperature of T0 = 302.15 K or 29°C. It ensures that the surface temperature matches the SST of 302.15 K. The constant zq1 determines the rate of decrease of the specific humidity with height and is set to 3000 m. In addition, the constant zq2 is set to 8000 m to quickly nudge the specific humidity profile toward 0 in the higher troposphere.
The analytic background virtual temperature sounding Tυ(z) is split into two different representations for the lower and upper atmosphere. The virtual temperature profile is given by
with the virtual temperature at the surface Tυ0 = T0(1 + 0.608q0) (approximately 306 K). The virtual temperature lapse rate Γ is set to 0.007 K m−1. It approximates the observed sounding from Jordan (1958) and is similar to the average lapse rate in the troposphere. Such a lapse rate provides conditionally unstable conditions in the troposphere. For simplicity, the virtual temperature in the upper atmosphere is set to the constant Tυt = Tυ0 − Γzt (approximately 201 K), which equals the virtual temperature at the tropopause height. As a result, the background temperature profile T(z) is
The analytic background vertical profiles of the temperature, specific humidity, and relative humidity are depicted in Fig. 1, along with a comparison to the soundings of Jordan (1958). The specific humidity values correspond to relative humidities of about 80% at lower levels and prescribe a warm and moist environment.
The background pressure profile p(z) is computed using the hydrostatic equation and ideal gas law. Note that the virtual temperature must be used in this calculation. As before, the background pressure profile has different representations in the lower and upper atmosphere:
where p0 = 1015 hPa is the background surface pressure, Rd = 287.04 J kg−1 K−1 is the ideal gas constant for dry air, and g = 9.806 16 m s−2 is the gravity. The pressure at the tropopause level zt is continuous and given by
which is approximately 130.5 hPa.
b. 2D axisymmetric vortex
The formulation of the 2D axisymmetric pressure field is inspired by an offline axisymmetric model, originally described in Nolan et al. (2001) and Nolan (2007). Here, we fit analytic functions to the pressure field from a similar offline model, which allows us to derive all other fields analytically. The fitted pressure equation p(r, z) comprises a background pressure profile p(z) [Eq. (4)] plus a 2D pressure perturbation p′(r, z), where r symbolizes the radial distance (or radius) to the center of the prescribed vortex. The chosen center position is further explained in section 2c and also listed in Table 1. The pressure is expressed as
The pressure perturbation depends on the pressure difference Δp between the background surface pressure p0 and the pressure at the center of the initial vortex, and is chosen to decay exponentially in radius and height. The quantity rp determines the pressure change in the radial direction and zp = 7 km prescribes how fast the pressure difference decays in height within the vortex. The pressure perturbation becomes negligible in the upper atmosphere and is set to zero above zt. Both Δp and rp determine the initial maximum wind and the initial RMW. The values are specified in section 4 for a variety of initial conditions.
In the limit of large r the expression approaches the background pressure profile p(z) as expected. The surface pressure ps(r) is computed by setting z = 0 m in Eq. (8), which gives
Such a representation of the surface pressure is similar to that derived for mature hurricanes by Holland (1980). It describes an idealized vortex with a pressure decrease of Δp in its center.
Next, we calculate an analytic function for the axisymmetric virtual temperature Tυ(r, z) using the hydrostatic equation and ideal gas law
which leads to the following expression:
It describes the warm-core structure of the initial vortex. The expression for the lower troposphere below zt approaches the background temperature profile [Eq. (2)] in the limit of large r, as expected.
The axisymmetric specific humidity q(r, z) is set to the background profile everywhere:
Therefore, the virtual temperature can simply be converted to the temperature T(r, z), resulting in the following expression:
Because of the small specific humidity value in the upper atmosphere (10−8 g kg−1 for z > zt) the virtual temperature equals the temperature to a very good approximation in this region. Equation (13) can also be expressed in the form of the background temperature T(z) [Eq. (3)] plus a temperature perturbation T ′(r, z):
Then the temperature perturbation is
In the limit of large r the temperature perturbation goes to zero.
Figure 2 shows the vertical temperature profile of the environmental conditions versus the temperature profile of an insulated parcel of air that is hypothetically lifted from the surface. The profiles are shown both for an environmental (unperturbed) background position and for the center of the vortex. The parcel first follows the dry adiabatic lapse rate. Once the parcel becomes saturated it cools at the moist adiabatic lapse rate and becomes buoyant. Therefore, the initial conditions are unstable for saturated air. The lifting condensation level (LCL) for the background environment is about 380 m and for the center of the vortex is about 400 m.
Finally, the tangential velocity field υT(r, z) of the axisymmetric vortex is defined by utilizing the gradient–wind balance, which depends on the pressure [Eq. (8)] and the virtual temperature [Eq. (11)]. The tangential velocity is given by
where fc = 2Ω sin(ϕc) is the Coriolis parameter at the constant latitude ϕc and Ω = 7.292 115 × 10−5 s−1 is the rotational speed of the earth. We set ϕc = π/18 (or 10°N), which is also the center latitude of the vortex for all experiments. Substituting Tυ(r, z) and p(r, z) into Eq. (16) gives
The tangential wind is zero in the upper atmosphere since the pressure no longer depends on the radial distance r above zt. Here, it is evident that the initial maximum wind speed υ0 = max|υT| and RMW are dependent on many different parameters, but the only free parameters are Δp and rp.
c. 3D spherical representation with height-based vertical coordinate
To initialize a global model with the idealized vortex, the axisymmetric fields need to be defined on the GCM grid. Here, we use spherical coordinates where λ and ϕ denote the longitudinal and latitudinal positions, respectively. The vertical coordinate is represented by the height above the surface. A coordinate transformation to other representations, such as Cartesian coordinates, is straightforward. We introduce the spherical grid by redefining r to be the great circle distance to the center of the vortex. The great circle distance is given by
where a = 6.371 22 × 106 m symbolizes the radius of the earth. The vortex is centered at (λc, ϕc) = (π, π/18), which corresponds to the position 10°N, 180°.
The great circle distance replaces r in the equations for the surface pressure [Eq. (9)], specific humidity [Eq. (12)], temperature [Eq. (13)], and the tangential velocity [Eq. (17)]. The surface geopotential Φs is set to zero. If nonhydrostatic model formulations are employed the vertical velocity needs to be set to zero. As mentioned before, the surface temperature in the background environment is identical to the prescribed SSTs of 29°C.
As a last step, the tangential velocity Eq. (17) needs to be split into its zonal and meridional wind components u(λ, ϕ, z) and υ(λ, ϕ, z) in spherical coordinates. Similar to Nair and Jablonowski (2008) this is done via the following expressions:
which are utilized in the following projections:
A small ε = 10−25 value avoids divisions by zero. The analytic initial conditions can easily be computed on any GCM grid provided the positions (λ, ϕ, z) are known. The vortex is well balanced on the spherical grid, but not in exact balance because of the use of the axisymmetric approach with constant Coriolis parameter. However, the balance is sufficient to foster the evolution of the preexisting, low-level vortex into an intense tropical cyclone–like vortex over a forecast period of 10 days. It provides a suitable basis for the studies of horizontal resolutions as discussed in sections 4 and 5.
d. 3D spherical representation with pressure-based vertical coordinate
For GCMs that are built upon pressure-based vertical coordinates, such as the σ coordinate (Phillips 1957) or the hybrid σ pressure so-called η coordinate (Simmons and Burridge 1981), the analytic initial conditions can be converted into the pressure-based systems. The conversion is analytic for the background conditions and in the upper atmosphere above zt, but requires straightforward fixed-point iterations in the vortex-covered region. Given the background pressure profile [Eq. (4)], where p0 is now generalized to be ps, the conversion between pressure p and height z is given by
The pressure p denotes the vertical pressure position of a GCM grid point, which can be computed via the known surface pressure [Eq. (9)] for σ or η coordinates. The corresponding z value can now be plugged into the equations for the specific humidity, temperature, and horizontal velocities according to Eqs. (12), (13), (22), and (23).
Note again that this analytic conversion is not accurate within the vortex because of the pressure perturbation and warm-core structure. Within the vortex the z value at each model level needs to be computed iteratively via Newton’s method:
The superscript n = 0, 1, 2, 3, … indicates the iteration count. The function F is determined by
Here pmodel is the pressure of the GCM grid point at a given longitude λ, latitude ϕ, and model level and p(λ, ϕ, z) represents Eq. (8) evaluated with the great circle distance r. Therefore, ∂F/∂z is defined by
which can be computed analytically from Eq. (8). The analytic form of ∂p/∂z in terms of the great circle distance r is
Equation (25) is iterated until it converges to |zn+1 − zn|/|zn+1| < ɛ, where ɛ is set to 2 × 10−13 (close to machine precision for double precision arithmetic). We recommend starting the iterations with the start value z0 equal to z given in Eq. (24). Typically, the computations converge within the ɛ precision in under 10 iterations. We apply the iterative technique below zt (equivalent to pmodel > pt) within a great circle distance of r ≤ 1000 km from the vortex center. It represents the distance at which the pressure [Eq. (7)] and temperature [Eq. (15)] perturbations become negligible.
It is possible to initialize the GCM without the iterative method using the background pressure to height conversion [Eq. (24)], but this introduces inaccuracies in the initial temperature and wind fields within the troposphere. We strongly recommend the iterative procedure to foster model intercomparisons between models with height-based and pressure-based vertical coordinates. As demonstrated in section 4c, the evolution of the vortex is sensitive to the initial conditions.
e. Characteristics of the initial conditions
Figure 3 shows the horizontal cross sections of the initial wind speed at a height of 100 m, the surface pressure, and the temperature at 4.35 km. The latter corresponds to the altitude of the maximum (warm core) temperature perturbation, which is about 3 K. The vortex is computed with the parameters rp = 282 km and Δp = 11.15 hPa, which leads to a maximum wind speed of υ0 of 20 m s−1 and an RMW of 250 km. The surface pressure is lowest in the center of the storm with a central pressure of 1003.85 hPa. In addition, the wind is greatest at the RMW and decreases rapidly in magnitude toward the center of the vortex, where the wind is zero. The wind field also decreases at radii larger than RMW and approaches the zero background flow at an approximate distance of about 1200 km from the center.
Figure 4 shows longitude–height cross sections of the magnitude of the wind, the pressure perturbation, the temperature perturbation, potential virtual temperature, square of the moist Brunt–Väisälä frequency, and the relative vorticity through the center latitude of the vortex. From the vertical cross section of the wind, we see that the wind is greatest at the surface and decays with height. The pressure perturbation is greatest at the surface and center of the vortex and decays with height and radius from the center, as expected from Eq. (7). The cross section of the temperature perturbation reiterates that the maximum perturbation occurs at a height of 4.35 km and decreases in magnitude radially and vertically, both below and above the maximum. The plot of the potential virtual temperature and the square of the moist Brunt–Väisälä both show that the initial vortex is stable to unsaturated air, but as indicated earlier in Fig. 2 is conditionally unstable with a lifting condensation level between 380 and 400 m throughout the vortex. The vorticity cross section displays that the maximum vorticity occurs at the center and decays radially and vertically until it reaches a minimum approximately 5° west and east of the center.
The maximum potential intensity (MPI) based on the theory of Emanuel (1986) is approximately 66 m s−1 for this environmental sounding. The MPI is calculated using the surface temperature of 29°C, or 302.15 K, assuming that the outflow temperature is equal to the tropopause temperature of 201 K, and that the ambient boundary layer relative humidity is 80%. In addition, the ratio of the boundary layer exchange coefficients Ck/CD is taken to be 1, r0 = 1200 km, pa = 1015 hPa, and f = 2Ω sin(ϕc) [for definitions of these variables see Emanuel (1986)]. During the simulation the ratio Ck/CD fluctuates. Therefore, our assumption that Ck/CD is 1 is an approximation that contributes uncertainty to the MPI estimate.
3. CAM 3.1 model description
We test and evaluate the effects of the initial conditions in idealized tropical cyclone simulations with the hydrostatic GCM CAM 3.1 (Collins et al. 2004, 2006). CAM is part of NCAR’s Community Climate System Model (CCSM) that is routinely used for climate change projections. We utilize a CAM 3.1 configuration with the mass-conservative finite-volume (FV) dynamical core in flux form (Lin 2004) that is built upon a 2D shallow-water approach in the horizontal plane. The vertical discretization follows a “Lagrangian control volume” principle, which is based on a terrain-following “floating” Lagrangian coordinate system and a fixed “Eulerian” reference frame. In particular, the vertically stacked finite volumes are allowed to float for a duration of several (in our simulations 10) short dynamics time steps before they are mapped back monotonically and conservatively to a fixed hybrid reference system. The physics parameterizations are called immediately after the vertical remapping step. The advection algorithm makes use of the monotonic Piecewise Parabolic Method (PPM) with an explicit time-stepping scheme. A regular latitude–longitude computational mesh is selected that includes both pole points. The prognostic variables are staggered as in the Arakawa-D grid. An almost identical FV dynamical core with different physics parameterizations and SSTs was also used in the tropical cyclone studies by Atlas et al. (2005), Shen et al. (2006a), and Zhao et al. (2009).
CAM 3.1 is run with the identical (Δλ, Δϕ) horizontal grid spacings of 1.0°, 0.5°, 0.25°, and 0.125° and 26 vertical η levels (L26). The hybrid coefficients for the standard CAM 3.1 vertical levels are documented in Jablonowski and Williamson (2006b). The four horizontal resolutions correspond to grid spacings of about 110, 55, 28, and 14 km in the equatorial region. The dynamics time steps at these 4 resolutions are 180, 90, 45, and 22.5 s, respectively. The model is run with the full CAM 3.1 physics parameterization suite and utilizes the aquaplanet setup as proposed by Neale and Hoskins (2000), but with constant sea surface temperatures of 29°C. These isothermal SSTs prescribe very warm ocean conditions and avoid latitudinal gradients in the initial background surface pressure or atmospheric temperature fields. The only external forcing is the distribution of the insolation at the top of the atmosphere. In particular, the solar irradiance is set to equinox conditions with a solar constant of 1365 W m−2. In addition, the distributions of atmospheric constituents, such as ozone, carbon dioxide, methane, and nitrous oxide are prescribed. The ozone distribution is zonally symmetric. The geophysical constants, including the earth’s rotation rate, the gravitational acceleration, and gas constants follow the suggestions by Neale and Hoskins (2000).
The CAM 3.1 physics suite is described in detail in Collins et al. (2004). It incorporates the Zhang and McFarlane (1995) deep convective parameterization, shallow moist convection and dry adiabatic adjustment, cloud microphysics, orographic gravity wave drag, the radiative effects of aerosols, and parameterizations of shortwave and longwave radiation. In addition, the bulk exchange formulations for surface fluxes are based on the Monin–Obukhov similarity theory, and the vertical diffusion and boundary layer processes with turbulent mixing depend on static stability indicators.
4. Sensitivity of the cyclone to initial conditions
In this section we provide a series of sensitivity tests to varying initial conditions, including the initial maximum wind speed and the RMW. For this study the RMW is assessed as the great circle distance between the wind maximum to the location of the vortex center as determined by the surface pressure minimum. Using the location of the surface pressure minimum as the vortex center can lead to errors of up to a half-grid spacing in the estimated RMW, which should be kept in mind during the analysis. In addition, we supply sensitivity tests to a different initial moisture profile and to small changes in the initial velocity and temperature fields. The aquaplanet tests are run for 10 days at both 0.5° and 0.25° horizontal resolutions. The results inform us about a configuration that triggers an intense tropical cyclone–like vortex from an initial vortex seed. A selected configuration is then used as the control case for further convergence studies in section 5.
a. 0.5° resolution sensitivity to initial size and strength
The first analysis assesses variance of the initial maximum wind speed while keeping the initial RMW constant at 250 km. Maximum wind speeds υ0 are 15, 20, 25, and 30 m s−1, which are derived from the prescribed rp and Δp values discussed in section 2. The corresponding rp and Δp values for the four wind values are listed in Table 2. Figure 5 shows the time evolution of the minimum surface pressure, maximum wind speed, and radius of maximum wind at 100 m for the four initial setups. The wind speed is linearly interpolated to this height level using the wind speeds and heights of the two surrounding model levels. The lowermost model level always lies below 100 m, which avoids extrapolation.
Figure 5 reveals that the initial vortex does not intensify immediately. The time it takes for the initial vortex to intensify seems to be dependent upon the initial intensity. The weakening shortly after the initialization is most likely due to the spindown of the vortex due to surface friction and the need for the development of a secondary circulation. The secondary circulation is absent in the initial conditions and takes some time to spin up. In addition, the vortex is not in perfect balance on the spherical grid as mentioned earlier. Both the surface pressure and maximum wind speed plots show that for the two strongest initial conditions, υ0 = 30 m s−1 and υ0 = 25 m s−1, there is an initial weakening of the vortex for the first day and first 2 days, respectively, after which point the vortex begins to intensify. For the υ0 = 20 m s−1 case we also see a slight weakening of the initial vortex, but although the vortex starts to intensify at day 2, it does not intensify significantly until day 6.
The results suggest that there is a threshold value of the initial strength of the vortex υ0 between 15 and 20 m s−1, below which development does not occur. This is evidenced by the υ0 = 15 m s−1 case, which never intensifies significantly within 10 days. In addition, further tests (not shown) reveal that an initial vortex with υ0 = 15 m s−1 never develops significantly for any initial RMW = 150–300 km.
The second group of tests explores varying initial values of the RMW from 150 to 300 km in increments of 50 km, while holding the initial maximum wind speed at a constant 20 m s−1. To keep the maximum wind speed constant at different RMW both rp and Δp have to be adjusted. The values are listed in Table 2. Note again that the discrete grid spacing in CAM 3.1 causes the initial RMW in the model to slightly deviate from the theoretical values.
The results in Fig. 6 show that the vortex intensity initially decreases for a day, at which point the 3 cases with the largest initial RMW (200–300 km) start to intensify. By day 2, the vortex for these three cases has intensified and begins a rapid intensification phase at day 6. The 2 largest RMW cases become the strongest, with the 250-km case showing maximum 100-m wind speeds of about 48 m s−1 at day 10. While the third case (RMW = 200 km) intensifies, it does not become as strong as the other 2 cases after 10 days. For the fourth case, RMW = 150 km, the initial vortex weakens for 2 days and remains rather constant in intensity until day 6. It thereafter begins to slightly intensify, but it does not actually pass its initial maximum wind speed of 20 m s−1 during the 10-day simulation.
When the initial vortex is weak, it apparently needs to be of an initial RMW equal to or greater than roughly 200 km in order to intensify. Further experiments at 0.5° (not shown) give evidence that an initial vortex can intensify when the initial RMW is less than 200 km, but to do so requires that υ0 be substantially greater than 20 m s−1 (≈30 m s−1, which is nearly hurricane strength). The fact that the 0.5° simulation can support 100-km vortices by day 10 demonstrates that the reason why smaller initial vortices fail to develop may not be simply an issue of inadequate horizontal resolution. However, further tests (described later) show that the RMW = 150 km case does develop at 0.25° resolution, so the failure mechanism is at least indirectly related to model resolution. Possible failure mechanisms here include the initial structure of the vortex (e.g., imbalance) and the nature of convection at lower horizontal resolutions. However, an in-depth investigation of these factors is not the focus of this study.
These results seem to contrast somewhat with those of Rotunno and Emanuel (1987) and Emanuel (1989). For example, Rotunno and Emanuel (1987) found that larger initial vortices struggle to develop. We find that an additional vortex simulation with an initial RMW of 500 km (not shown) is only slightly weaker by day 10 than those with smaller initial RMW. However, there is no clear relationship between the initial RMW and the later strength for the smaller initial vortices in Fig. 6 (omitting the RMW = 150 km case, which does not develop). In addition, the somewhat weaker evolution for the initial RMW = 200 km case might reflect the fact that the vortex lies in the transition region wherein the structure of the smaller initial vortices cannot be supported by the model. Emanuel (1989) also found that the length scale of the mature vortex depends on that of the initial vortex. In contrast, we find that there is no relationship between the initial and final RMW. Furthermore, we find that varying the initial RMW has less of an impact on the final RMW variability than does varying the initial maximum wind speed υ0 (Fig. 5). A better understanding of how our results compare with those two studies would require additional simulations that are beyond the scope of this study. Nonetheless, any differences from Rotunno and Emanuel (1987) and Emanuel (1989) are seen as a result of a differing initial vortex structure and the use of a different model with varying model resolution and model physics.
From the sensitivity studies of the varying initial conditions we choose a control case for further analysis. The control case produces the strongest storm from a weak initial vortex. We choose the case with υ0 = 20 m s−1 and RMW =250 km as our control case, which corresponds to rp = 282 km and Δp = 11.15 hPa. Figure 7 shows the intensification of the wind speed for the control case, with specific snapshots at days 0, 5, and 10. The top row of Fig. 7 displays the horizontal cross section of the magnitude of the wind at 100 m. The bottom row of Fig. 7 shows the longitude–height cross section of the magnitude of the wind through the center latitude of the vortex. The time series displays the intensification of the vortex from an initial surface vortex to a tropical cyclone–like vortex. We also see strong vertical development, especially near the RMW. During the entire simulation the cyclone experiences beta-drift toward the northwest Holland (1983). There is also somewhat of a resemblance of a calm eye that forms as the vortex intensifies. However, because of resolution constraints the eye is not completely defined. The cyclone is a warm-core vortex (not shown).
b. 0.25° resolution sensitivity to initial size and strength
We repeat the sensitivity tests at a higher horizontal resolution of 0.25°, or about 28 km, to assess the robustness of the sensitivity analysis and the control case. Again, we review the sensitivity to the varying initial maximum wind speeds with a constant RMW of 250 km presented in Table 2. Similar to the 0.5° tests depicted in Fig. 5, Fig. 8 shows that at first the intensity weakens, and the initially stronger vortices rebound from this weakening more quickly and strongly. Rapid intensification occurs after 1 day for the strongest case, υ0 = 30 m s−1, and at days 2 and 3 for the υ0 = 25 m s−1 and υ0 = 20 m s−1 cases, respectively. The υ0 = 15 m s−1 case intensifies substantially more within the 10-day simulation (but the intensity is still less than the other 3 cases) at the 0.25° resolution than that at the 0.5° resolution, reaching an intensity of over 2.5 times that of the lower resolution. This suggests that the threshold intensity needed to foster development and rapid intensification is resolution dependent.
All three plots in Fig. 8 show that the vortices develop into more intense and more compact cyclones at the higher resolution, with the maximum wind speed at 100 m approaching a value of 62 m s−1 for the strongest three cases. Such a storm would correspond to a category-4 hurricane on the Saffir–Simpson scale. Additionally, the RMW for each case seems to converge toward a value in between roughly 50 and 80 km at day 10 for all cases, which is smaller than the RMW range at the 0.5° resolution. Similar resolution dependencies are also discussed in Bengtsson et al. (2007) and Hill and Lackmann (2009).
The second set of tests at the 0.25° resolution is an examination of the sensitivity to varying initial RMW with a constant maximum wind speed of υ0 = 20 m s−1. The associated rp and Δp values are again listed in Table 2. The results are very similar to those with the 0.5° grid spacing. Initially, the intensity of all vortices weakens for 2 days and then increase as seen in Fig. 9. However, the vortices begin rapid intensification by day 3, which is 3 days earlier than for the 0.5° tests (Fig. 6). By day 10 the maximum 100-m wind speed approaches 62 m s−1 for the two cases with the largest RMW. In addition, the case with an initial RMW of 150 km develops substantially more at 0.25° than at 0.5° resolution, indicating that a vortex with a smaller RMW more readily develops when the model resolution is higher.
The RMW plot in Fig. 9 shows that as the vortex develops the RMW ranges between 50 and 100 km in all cases. They have a smaller RMW at day 10 as compared to the 0.5° resolution runs. Similar to the 0.5° resolution runs, varying the initial maximum wind speed has more of an effect on the subsequent RMW variability than does varying the initial RMW. Given a suitable initial size the initial maximum wind speed υ0 of the cyclone seems to be the decisive factor that determines the potential for development and rapid intensification.
The analysis suggests that the control case described above will suffice in the simulation of the growth of the initial vortex into a tropical cyclone for both the 0.25° and 0.5° simulations. Figure 10 is the same as Fig. 7, but for 0.25° resolution. It displays snapshots of the wind speed for the control case in both a horizontal and vertical cross section at days 0, 5, and 10. At the higher resolution the cyclone becomes substantially (about 14 m s−1) stronger by day 10. The day-5 snapshot also shows that the storm has intensified quicker than the 0.5° resolution case (Fig. 7). In addition, there is more significant vertical development, with a noticeable slant of the eyewall, and the storm is more compact as indicated by a smaller RMW.
c. 0.25° resolution sensitivity to small changes in the initial fields
The model sensitivity to tiny changes in the initial temperature and velocity fields within the vortex is tested. This is done by changing the value ɛ to which the iterations in Eq. (25) converge. This produces very small changes in the height and therefore initial fields within the vortex. The model is run for 3 values of ɛ that are 2 × 10−13, 2 × 10−12, and 2 × 10−11. These small changes in ɛ result in maximum absolute differences in the zonal and meridional velocity fields of about 1 × 10−13 m s−1 and in the temperature field of about 1 × 10−12 K. Figure 11 displays the time evolution of the minimum surface pressure, maximum wind speed, and radius of maximum wind at 100 m for the three different values of ɛ using the control specific humidity at the surface (q0 = 21 g kg−1). It is evident that even very small changes in the model initialization can cause some small but notable variations in the storm’s intensity and size during the 10-day simulation. This indicates that there might be a predictability limit after which the solutions have greater uncertainty. Such a limit of predictability due to minute changes in initial conditions is also observed in Zhang and Sippel (2009) using a mesoscale model and the study implies that the predictability of tropical cyclones in models is restrained at all time scales.
d. 0.25° resolution sensitivity to the moisture profile
We also test the sensitivity to the low-level moisture. In Eq. (1) we set the specific humidity at the surface q0 to 21 g kg−1 according to the Jordan (1958) relative humidity profile with the surface temperature of 29°C. However, q0 could also be set to the specific humidity value of the Jordan (1958) sounding at the surface, which is 18.5 g kg−1. This would cause a change in the initial moisture profile and therefore the initial temperature profile. It is informative to evaluate the sensitivity of the model simulations to such variations of the initial conditions as shown by the fourth experiment in Fig. 11. The time evolution of the minimum surface pressure, maximum wind speed, and radius of maximum wind at 100 m can be compared for the two cases with ɛ = 2 × 10−13 that utilize the default setting q0 = 21 g kg−1 and the new value of 18.5 g kg−1. It is apparent that the change in the initial profiles causes only small changes, on par with those discussed earlier in section 4c. The q0 = 18.5 g kg−1 case with decreased low-level moisture results in a slightly weaker storm at day 10, but the variation lies within the uncertainty range discussed above. This small sensitivity to the initial moisture profiles could be due to the use of a relatively coarse model resolution and the specifics in the CAM 3.1 model physics, in particular in the convection parameterization.
5. Horizontal resolution convergence test
The previous section identified the control case with υ0 = 20 m s−1 and RMW = 250 (rp = 282 km, Δp = 11.15 hPa). We now initialize this vortex at the wide range of horizontal resolutions (1.0°, 0.5°, 0.25°, and 0.125°) with 26 levels to gain insight into the physical characteristics of the idealized cyclone. All simulations are run with identical physical parameterizations without retuning the physics parameter set. The adjustable parameters have been derived for CAM 3.1 climate simulations with the Eulerian spectral transform dynamical core at the T85L26 resolution (see Collins et al. 2004). We use these defaults to match the CAM 3.1 aquaplanet setups assessed in Williamson (2008a,b). We do not change the vertical resolution since the CAM 3.1 physics parameterization suite is known to be sensitive to the placement of the levels.
Figure 12 presents the time evolution of the minimum surface pressure, maximum wind speed at 100 m, and the position of the storm centers at all four horizontal resolutions. The storm centers are determined by the gridpoint locations of the minimum surface pressure. The filled circles in Fig. 12c denote the daily positions over the 10-day simulation period.
As the horizontal resolution increases the vortex becomes more intense by day 10. The maximum 100-m wind at day 10 is 48.7 m s−1, 63.0 m s−1, and 72.4 m s−1 for the horizontal resolutions of 0.5°, 0.25°, and 0.125°, respectively. Note that the absolute maximum wind occurs at approximate heights of 1–1.5 km, where wind speeds are about 6–11 m s−1 greater than those at the height of 100 m. The storms simulated at the higher resolutions also intensify earlier during the simulation, as seen in the minimum surface pressure and maximum wind speed plots. This is evidenced by the fact that after the initial weakening it takes the vortex 132, 84, and 72 h at 0.5°, 0.25°, and 0.125°, respectively, to surpass the initial maximum wind speed of 20 m s−1. In the 1.0° simulation the storm never becomes as strong as the initial vortex. There is no sign of convergence in intensity at the highest resolutions, which might suggest that even higher resolutions are desirable.
In addition, Fig. 12c shows that the horizontal resolution impacts the storm track and location of the storm center. The spread in the positions increases notably after day 8. However, the center positions do not change systematically with resolution.
When simulated at 0.125° (Fig. 13), the cyclone is stronger and more compact than the storms at both 0.5° (Fig. 7) and 0.25° (Fig. 10) simulations. Similar results were also found by Bengtsson et al. (2007). The 0.125° cyclone has a distinct slanted eyewall-like structure and relatively calm eye at day 10. The 100-m maximum wind speed at day 10 of 72.4 m s−1 is equivalent to a category-5 hurricane on the Saffir–Simpson scale.
It is interesting to note that this intensity is larger than the estimated Emanuel’s MPI of about 66 m s−1, which could be related to inadequate physics parameterizations at these high resolutions. As mentioned before, no tuning of the physics parameterizations, such as the convection and boundary layer parameterizations, were made. It is possible that this allows for storms that are too intense. However, further investigations of the physics parameterizations are beyond the scope of this investigation and will be a subject of future work. Simpler axisymmetric hurricane models and full-physics 3D models have also been documented to produce storms that exceed Emanuel’s MPI. This is discussed in Persing and Montgomery (2003), Bryan and Rotunno (2009), and Wang and Xu (2010).
Figure 14 displays the precipitation rate for large-scale, convective, and the total precipitation at day 10 for the 0.5°, 0.25°, and 0.125° simulations. It can be seen that as the horizontal resolution increases from 0.5° to 0.125° the precipitation fields show rainband-like features spiraling out from the storms center that become increasingly resolved. These rainband-like features are apparent in both the large-scale and convective precipitation fields.
Figure 14 also shows that for all simulations the large-scale precipitation contributes the majority to the total precipitation near the central core of the storm. Even though Fig. 13 shows a relatively calm eye there is still precipitation in the eye at the 0.125° resolution. The most intense total precipitation rate for the 0.125° simulation is near the center of the storm and has a maximum value of about 88.22 mm h−1, which represents a 6-h average. This rather extreme peak precipitation rate only covers a very small region, but could provide another hint that the physics parameterizations might need retuning at these high resolutions with hurricane-strength winds.
Table 3 lists the maximum large-scale, convective, and total precipitation rates at days 5 and 10 for the 0.5°, 0.25°, and 0.125° simulations. As the resolution increases the maximum 6-h average total and large-scale precipitation rates increase, while the maximum convective precipitation rate decreases mainly in the core of the storm. These results are similar to other aquaplanet studies that address the convergence with resolution aspects of global precipitation (Williamson 2008b) and equatorial precipitation fields (Lorant and Royer 2001).
As a last point we assess the realism of highest resolution (0.125°) simulation used in the convergence study. Figure 15 displays vertical cross sections through the center latitude of the vortex as a function of the radius from the center of the vortex for the temperature, relative humidity, and vertical pressure velocity at days 0, 5, and 10 for the 0.125° simulation. The top row shows the evolution of the warm core. At day 0 there is only a slight perturbation representing the warm core, and by days 5 and 10 the perturbation has become more defined.
The middle row of Fig. 15 shows the relative humidity. At day 0 the maximum is located at the surface as prescribed in the initial conditions. By day 5 the troposphere near the center of the storm has moistened, reaching saturation in some locations. By day 10 the atmosphere has moistened even more, especially near the eyewall at the radius of about 50 km, with high relative humidities reaching high into the troposphere. In addition, the day-10 relative humidity field is rather symmetric about the center of the storm, indicating an organized, intense storm. Another indication of an organized storm is the relative humidity minimum located at the center of the storm just below the tropopause. The large values of the relative humidity at the center of the storm in the lower and midtroposphere are linked to the precipitation that is seen at the center (Fig. 14). At day 10 the relative humidity field also shows hints of spiral rainbands with areas of maximum relative humidity that extend into the vertical at a radius slightly larger than 200 km.
The vertical pressure velocity fields (bottom row of Fig. 15) confirm the evolution of the intense organized storm by day 10, with a maximum value of about −15.72 Pa s−1 near the RMW as expected with intense updrafts in the eyewall. At day 10, there is also a distinct downdraft region slightly (25 km) west of the storm center positioned at (28.75°N, 165.25°E). The downdrafts reach from the surface to a height of about 9 km. Note that initially (day 0) the pressure velocity is 0 and therefore omitted in the figure.
Three conclusions can be drawn from the convergence study. First, the vortex seed as specified in the control case develops into a strong tropical cyclone–like storm at all resolutions except at the coarest 1° grid spacing. Second, the physical characteristics of the idealized storms have many realistic features, especially the highest resolution (0.125°) simulation. Third, the model simulations do not converge in intensity in the range of horizontal resolutions between 110 and 14 km. The latter aspect needs further investigation. Overall, the results suggest that the analytic functions represent a robust initialization technique to study idealized cyclones in GCMs.
6. Summary and conclusions
The paper introduces an analytic initialization technique for idealized tropical cyclone experiments in atmospheric general circulation models. The initial axisymmetric vortex is in hydrostatic and gradient–wind balance and provides favorable conditions for rapid intensification over a simulation period of 10 days. The initial dataset triggers the evolution of a single, weak warm-core vortex into a much stronger tropical cyclone–like storm. This has been demonstrated in high-resolution simulations with the Community Atmosphere Model version 3.1 (CAM 3.1). The simulations were conducted in an aquaplanet configuration with constant SSTs of 29°C. Ten-day model simulations with grid spacings of 1°, 0.5°, 0.25°, and 0.125° and 26 vertical levels were shown.
Several sensitivity studies were presented that highlighted the effects of varying initial conditions. In particular, the role of the maximum initial wind speed and radius of maximum wind has been investigated. We showed that the initial vortex must satisfy certain threshold conditions to support the intensification of the preexisting vortex. We found that the prescribed vortices with a maximum initial wind speed of 15 m s−1 did not develop into strong tropical storms in both the 0.5° and 0.25° simulations during the 10-day assessment period. A suitable vortex must therefore exhibit a maximum initial wind speed that is stronger than this threshold, approximately 20 m s−1, for the described analytic initialization technique. In addition, the initial RMW needs to be at least 200–250 km wide to foster the intensification of the preexisting storm that is initially weak (υ0 = 20 m s−1). However, the model can support storms with RMW of 100 km and this initial RMW threshold of 200–250 km is possibly related to the initial structure of the vortex and the behavior of the model physics. These thresholds do vary with model resolution. In general, the high-resolution experiments with 0.25° grid spacing lead to more intense and compact storms than the lower resolution (0.5°) simulations.
A favorable initial configuration was picked as a candidate to study the impact of horizontal resolution on the evolution of the tropical cyclone–like vortices. This control case has an RMW of about 250 km and a 20 m s−1 maximum initial wind speed and is used as the basis for a second group of sensitivity tests in which the horizontal resolution is recursively halved from 1° to 0.125°. As the resolution increases the initial vortex produces storms that are more intense and compact. The storm simulated at the finest 0.125° resolution even exceeds Emanuel’s maximum potential intensity limit of about 66 m s−1. This gives an indication that the physics suite might need to be modified at high resolutions. At the highest resolution the model starts to resolve rainband-like structures as seen from the precipitation and relative humidity fields. In addition, we see regions of intense updrafts near the RMW. However, despite the relatively calm eye (in terms of wind speed) there is still precipitation and large relative humidity values within the eye. Again, this might indicate potential inadequacies within the physics package at high resolutions or for intense hurricane conditions. It also could indicate that even higher resolutions are necessary, both in the horizontal and, equally important, vertical directions.
In future work, we will use these idealized initial conditions to assess different types of physics parameterizations and their impact on tropical cyclone development in GCMs. In addition, we plan to assess the impact of different dynamical cores on the evolution of idealized cyclones.
The authors thank Jerry Olson (NCAR) for providing us with the aquaplanet version of CAM3.1 and giving us advice on the model configuration. We also thank David Nolan (University of Miami) for providing us with the axisymmetric model that inspired the development of the analytic initial conditions. We would like to acknowledge the high-performance computing support provided by NCAR’s Computational and Information Systems Laboratory, which is sponsored by the National Science Foundation. We thank the reviewers for their helpful suggestions.
Corresponding author address: Kevin Reed, University of Michigan, 2455 Hayward St., Ann Arbor, MI 48109. Email: firstname.lastname@example.org